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%I #19 Oct 09 2018 15:10:53
%S 1,1,1,1,1,2,4,1,1,1,3,9,5,7,1,1,1,4,16,14,26,8,10,1,1,1,5,25,30,70,
%T 34,52,11,13,1,1,1,6,36,55,155,104,190,63,87,14,16,1,1,1,7,49,91,301,
%U 259,553,253,403,101,131,17,19,1,1,1,8,64,140,532,560
%N Triangle read by rows: T(j,0)=1, T(0,j) = [j=0], T(-1,j)=T(-2,j)=0, T(j,k)=2T(j-1,k)-T(j-2,k)+T(j-1,k-2)+T(j-2,k-2).
%H Lars Blomberg, <a href="/A247645/b247645.txt">Table of n, a(n) for n = 0..1680 (rows 0-40)</a>
%H Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%e Triangle begins:
%e 1,
%e 1,1,1,
%e 1,2,4,1,1,
%e 1,3,9,5,7,1,1,
%e 1,4,16,14,26,8,10,1,1,
%e 1,5,25,30,70,34,52,11,13,1,1,
%e 1,6,36,55,155,104,190,63,87,14,16,1,1
%e 1,7,49,91,301,259,553,253,403,101,131,17,19,1,1
%e 1,8,64,140,532,560,1372,806,1462,504,736,148,184,20,22,1,1
%e ...
%t T[_, 0] = 1; T[j_, k_] /; 0 <= k <= 2j := T[j, k] = Which[k<0 || k>2j, 0, k == 2j || k == 2j-1, 1, OddQ[k], T[j-1, k] + T[j-1, k-1], EvenQ[k], T[j-1, k-2] + 2 T[j-1, k-1] + T[j-1, k]];
%t Table[T[j, k], {j, 0, 8}, {k, 0, 2j}] // Flatten (* _Jean-François Alcover_, Oct 09 2018 *)
%Y Column 0-5 is A000012, A000027, A000290, A000330, A006325, A033455.
%K nonn,tabf
%O 0,6
%A _N. J. A. Sloane_, Sep 23 2014
%E More terms from _Lars Blomberg_, Aug 05 2015
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